{
  "id": "1502.04940",
  "version": "v1",
  "published": "2015-02-17T16:00:27.000Z",
  "updated": "2015-02-17T16:00:27.000Z",
  "title": "Stochastic Averaging in Discrete Time and Its Applications to Extremum Seeking",
  "authors": [
    "Shu-Jun Liu",
    "Miroslav Krstic"
  ],
  "comment": "32 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "We investigate stochastic averaging theory for locally Lipschitz discrete-time nonlinear systems with stochastic perturbation and its applications to convergence analysis of discrete-time stochastic extremum seeking algorithms. Firstly, by defining two average systems (one is continuous time, the other is discrete time), we develop discrete-time stochastic averaging theorem for locally Lipschitz nonlinear systems with stochastic perturbation. Our results only need some simple and applicable conditions, which are easy to verify, and remove a significant restriction present in existing results: global Lipschitzness of the nonlinear vector field. Secondly, we provide a discrete-time stochastic extremum seeking algorithm for a static map, in which measurement noise is considered and an ergodic discrete-time stochastic process is used as the excitation signal. Finally, for discrete-time nonlinear dynamical systems, in which the output equilibrium map has an extremum, we present a discrete-time stochastic extremum seeking scheme and, with a singular perturbation reduction, we prove the stability of the reduced system. Compared with classical stochastic approximation methods, while the convergence that we prove is in a weaker sense, the conditions of the algorithm are easy to verify and no requirements (e.g., boundedness) are imposed on the algorithm itself.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-17T16:00:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic averaging",
      "discrete time",
      "discrete-time stochastic extremum seeking algorithm",
      "lipschitz discrete-time nonlinear systems",
      "stochastic extremum seeking scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}